Fourier transform discovers periodicity in the source data, so applying it to some data points with nearly periodic patterns can often produce interesting results. For example, see Fourier transform of the Hilbert curve images. Still, the 3-dimensional look of the resulting image is something totally unexpected for me. Below is the same image, but rendered for a bigger fragment of the plane, 2048x2048.

Extensions

The idea can be extended by building the co-prime matrix not for integers but for some general integer sequence. This gallery contains some of the spectra, obtained for different sequences: Fibonacci numbers, \(n^2+1\), \(n^3+3\), \(n^2-n+41\) (Euler polynomial with many primes), partition function values and tribonacci numbers. Curious fact: the spectrum for the Fibonacci numbers is very similar visually to the original integers spectrum. But there is a difference, the image below shows two spectra superimposed, with yellow color for integers, and blue color for Fibonacci numbers.

Source code

The images above were generated by a simple Python script below. See it here, if the gist fails to load. To run it, you will need to install Numpy and PIL (Python Imaging Library).
The code itself is fairly straightforward, in fact, the most complicated part is the adaptive calculation of the brightness diapason.